F p p Vxx yy f zz

The underwater pinger carries a high precision clock that is synchronized with those of the sonobuoys with DGPS prior to AUV deployment. Assuming the measurement s of the oneway travel time of the emitted signal by the pinger of the underwater vehicle is available to the ith sonobuoy, the distances between the AUV and the sonobuoys can be simply computed as f = V c( z) (2)

The sound speed is not a constant and varies with depth of the sea. Practically the weighted average speed is used to evaluate the sound speed propagating under the sea where the profile of the sound speed is divided as m layers, is shown on fig. 5. Then the sound speed considered in the central control ASV can be defined as c( z) = -k £(c.+ c+i) • (z+1 - Z)/2 (3)

Assuming the measurement performed each time affected by a bounded error ei, which accounts for all the uncertainties, such as synchronization errors, noisy signals, DGPS accuracy, buoy/vehicle motion between transmission and reception of the same ping, multi-path effect, etc. Then the observation equations with uncertainties can be written as

As the Taylor-series estimation (also Gauss or Gauss-Newton interpolation) gives a Least Squared error solution to a set of simultaneous nonlinear algebraic position equations (Foy, 1976). If Xg, yg, Zg are initial guesses of the true vehicle position, fi can be expanded in a Taylor-series keeping only terms below second order as df df df f + J-g Sx + Sy + Jjl Sz = s,c(z) - e, (5)

dx dy dz

Where  